The discussion revolves around a trigonometric identity involving the expression (sin^2x + 4sinx + 4) / (sinx + 2) and its equivalence to sinx + 2. Participants are exploring the validity of this identity and the implications of factoring the numerator.
Several participants have provided guidance on factoring and simplifying the expression. There is an ongoing exploration of the implications of canceling terms and the conditions that affect the validity of such operations. Questions remain about specific values of x that may affect the equality.
Participants are discussing the implications of specific values that could lead to undefined expressions, particularly focusing on the denominator and its role in the overall identity.
Please enclose the entirety of any numerator and/or denominator in parentheses.Veronica_Oles said:Homework Statement
sin^2x + 4sinx +4 / sinx + 2 = sinx +2
Homework Equations
The Attempt at a Solution
L.S = sin^2x + 4sinx +4 / sinx + 2
=1-cos^2+4(sinx + 1) / sinx +2
Not sure where to go from there.
Not sure if I was even supposed to factor out the 4?
SammyS said:Please enclose the entirety of any numerator and/or denominator in parentheses.
Factor the numerator.Veronica_Oles said:(Sin^2x + 4sinx + 4) / (sinx + 2) = sinx + 2
Thank you, didn't catch that.SammyS said:Factor the numerator.
So, what do you get ?Veronica_Oles said:Thank you, didn't catch that.
((Sinx + 2)(Sinx + 2)) / (Sinx + 2)SammyS said:So, what do you get ?
Veronica_Oles said:((Sinx + 2)(Sinx + 2)) / (Sinx + 2)
Then you cancel one from top and bottom to get: Sinx + 2.
micromass said:Yes, but it is a tiny bit more complicated. Here's something to think about:
1) Why doesn't the following equality hold for all ##x##:
\frac{(x+2)(x+2)}{x+2} = x+2
2) Why is this no problem with the question in the OP?
micromass said:Yes, but it is a tiny bit more complicated. Here's something to think about:
1) Why doesn't the following equality hold for all ##x##:
\frac{(x+2)(x+2)}{x+2} = x+2
2) Why is this no problem with the question in the OP?
micromass said:Yes, but it is a tiny bit more complicated. Here's something to think about:
1) Why doesn't the following equality hold for all ##x##:
\frac{(x+2)(x+2)}{x+2} = x+2
2) Why is this no problem with the question in the OP?
micromass asked two questions. You didn't respond to his first question, and your answer to the second question doesn't address why ##\frac{(\sin x+2)(\sin x+2)}{\sin x+2} = \sin x+2## is always true, regardless of the value of x.Veronica Oles said:((Sinx + 2)(Sinx + 2)) you then take reciprocal of denominator and multiply it by the numerator, and that it is when you cancel them out?